What a strange way to look at it, when most of the inhabitants of Middle Earth had no choice in the matter of whether they were involved in the business of the Ring, as catastrophe came to them regardless as to their interest, as evidenced by the return to the Shire at the end of the trilogy, the ending left out of the movies. Frodo even laments his own role at one point, and says he wishes that he did not live in a time with these events, and Gandalf assures him that is how all who live through historic events feel.

It was only because of the selfless sacrifice of Frodo and Sam that any may have been able to live a life untouched by the turmoil of the reawakening of Sauron. If they had failed, nobody would have had a choice whether or not to "care" about the Ring -- they'd simply be oppressed by it.